what is the Convergence radius and what happen at the edges?

Question

What is the Convergence radius and what happen at the edges?

$$\sum_{n=1}^{\infty}\frac{(x+2)^{n^2}}{n^n}$$

Thank you

Answer 1

We have (using the root test)

$$\left|\frac{(x+2)^{n^2}}{n^n}\right|^{1/n}=\frac{|x+2|^n}{n}\xrightarrow{n\to\infty}\ell<1\iff |x+2|\le 1\iff x\in[-3,-1]\;\text{so }\; R=1$$

Answer 2

As stated in a previous answer, you get radius of convergence equal to $1$ around $x = -2$. At the endpoints, you get an absolutely convergent series because $\sum_{n=2}^\infty 1/n^n < \sum_{n=2}^\infty 1/2^n$ which is a convergent geometric series so the series converges at the endpoints.